Unfinished work in queueing system with the input stream diffusion intensity with zero ratio of drift

An analytical model of information networks and their separate elements is the queueing system (QS). In this work, we construct a mathematical model of a QS as a system of equations for nonstationary and stationary characteristics of unfinished work in the QS. The QS is considered with one servicing device, exponential service, and infinite storage capacity. On the input, a doubly stochastic Poisson stream of requests with the diffusion intensity $\lambda(t)\in[\alpha,\beta]$ with springy boundaries is received. The diffusion process $\lambda(t)$ has a zero ratio of drift $a=0$ and diffusion coefficient $b>0$. The service time $\eta$ has arbitrary distribution with the distribution function $B(x)$.
The goal of this work is derivation of equations concerning the joint distributions of unfinished work and the intensity of the input flow in non-stationary and stationary modes.
The Kolmogorov dynamics is applied for obtaining equations on the characteristics of requests of unfinished work. Theorem 1 provides the equations in the case of non-stationary distribution of an unfinished work in QS with the transient regime. The initial and boundary conditions and equations for interior and boundary points are obtained. The equations are derived with the use of the semi-Markov process approximating the diffusion process. We show that the diffusion process with the zero coefficient of drift $a=0$ and diffusion coefficient $b>0$ is received from semi-Markov process as a result of limit transition. Theorem 2 gives the equations in the case of stationary distribution of an unfinished work in QS for a stationary mode. Boundary conditions are obtained.